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- How is Calculus Used in Everyday Life? | Toppr Bytes
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- Mathematics Applied to Physics and Engineering
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How is Calculus Used in Everyday Life? | Toppr Bytes
Notice that this formula now contains only one unknown variable. We are interested in maximising the area of the garden, so we differentiate to get the following:. The quantity that is to be minimised or maximised must be expressed in terms of only one variable. To find the optimised solution we need to determine the derivative and we only know how to differentiate with respect to one variable more complex rules for differentiation are studied at university level.
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We get the following two equations:. A wooden block is made as shown in the diagram. The diagram shows the plan for a verandah which is to be built on the corner of a cottage. The container has a specially designed top that folds to close the container. The cardboard needed to fold the top of the container is twice the cardboard needed for the base, which only needs a single layer of cardboard.
Mathematics Applied to Physics and Engineering
It is very useful to determine how fast the rate at which things are changing. Mathematically we can represent change in different ways. For example we can use algebraic formulae or graphs. Graphs give a visual representation of the rate at which the function values change as the independent input variable changes. This rate of change is described by the gradient of the graph and can therefore be determined by calculating the derivative.
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We have learnt how to determine the average gradient of a curve and how to determine the gradient of a curve at a given point. These concepts are also referred to as the average rate of change and the instantaneous rate of change. When we mention rate of change, the instantaneous rate of change the derivative is implied.
When average rate of change is required, it will be specifically referred to as average rate of change. Acceleration is the change in velocity for a corresponding change in time. Therefore, acceleration is the derivative of velocity.
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Determine the following:. Notice that the sign of the velocity is negative which means that the ball is moving downward a positive velocity is used for upwards motion. Just because gravity is constant does not mean we should necessarily think of acceleration as a constant.
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We should still consider it a function. A pump is connected to a water reservoir.
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Interpretation: this is the stationary point, where the derivative is zero. The ball has stopped going up and is about to begin its descent. We know that velocity is the rate of change of displacement. We also know that acceleration is the rate of change of velocity. The coefficient is negative and therefore the function must have a maximum value. The main approach and style of this book is informal, theorem-free and practical. By using an informal and theorem-free approach, all fundamental mathematics topics required for engineering are covered, and readers can gain such basic knowledge of all important topics without worrying about rigorous often boring proofs.
Some rigorous proof and derivatives will be presented in an informal way by direct, straightforward mathematical operations and calculations, and the students will gain the same level of fundamental knowledge without any tedious steps. In addition, the practical approach provides over worked examples in this book so that students can see how each step of the mathematical problems can be derivative without any gap or jump in steps.
Thus, readers can build their understanding and mathematical confidence gradually and in a step-by-step manner. Show more Show less. A compact and highly accessible introduction to many of the most pertinent areas of applied mathematics relevant to the engineering profession.